The criterion of information-theoretic stationarity diagnostics for the Monte Carlo simulation of nuclear criticality has been extended to undersampling diagnostics. Here, undersampling diagnostics means the posterior checking of the number of neutron histories per cycle. A statistically sound criterion using Shannon and relative entropies is defined based on the inequality with a penalty term for the minimum descriptive length of instantaneously decodable encoding. An alternative criterion based on a large sample property of particle population is defined within the information-theoretic framework of the asymptotic equipartition property and the method of types. An auxiliary criterion is proposed using the concave property of Shannon entropy. Numerical results are presented for the "k-effective of the world" problem by Whitesides. The results indicate that the estimation bias of the neutron effective multiplication factor will be reduced to a practically negligible level if these criteria are satisfied. It can be concluded that equilibrium is a stronger condition than stationarity concerning the source distribution in the Monte Carlo simulation.